The \(\gamma \) -SMI posterior in (REF ) is attractive as a formally straightforward family of candidate posteriors encompassing Cut models and Bayesian inference. However it is very awkward computationally and in fact we have no idea how to implement it in practice.
We now give two alternative interpolating sequences of candidate posterior distributions. The first is \(\eta \) -SMI, given in [1]}. We begin by introducing an auxiliary parameter \(\tilde{\theta }\) , expanding the model parameters from \((\varphi ,\theta )\)  to \((\varphi ,\tilde{\theta },\theta )\) .
The \(\eta \) -SMI posterior is
\( {\tilde{p}}^{(s)}_{\eta }(\varphi ,\tilde{\theta },\theta |Y, Z) = {\tilde{p}}^{(s)}_{\eta }(\varphi ,\tilde{\theta }|Y, Z) \pi (\theta |Y,\varphi )\) 
